Graphs: Graph of Linear Equation – JSS2 Mathematics Lesson Note (Second Term)

A linear equation is an equation whose graph is a straight line.

Behavioural objectives

By the end of the lesson, students should be able to:

1. state the slope of a horizontal line and explain why a vertical line has no slope;
2. rewrite a linear equation in “y =” form before graphing it;
3. check whether a point lies on a line by putting its coordinates into the equation;
4. make a table of values and use it to graph a linear equation;
5. name the origin, the abscissa and the ordinate on a coordinate plane.

What are the equations of straight lines?

A straight line can be written in four main ways: slope-intercept form, point-slope form, horizontal-line form and vertical-line form.

What is the slope of a line?

The slope of a line is a number that tells how steep the line is.

• Rise: how far the line goes up or down between two points.
• Run: how far the line goes across between the same two points.
• Slope: rise divided by run.
• Y-intercept: the point where the line crosses the y-axis, which is where x = 0.

What is the coordinate plane?

The coordinate plane is a flat surface made by a horizontal axis (the x-axis) and a vertical axis (the y-axis).

What is the origin?

The origin is the point (0, 0) where the x-axis and the y-axis cross.

• The two axes divide the plane into four regions called quadrants.
• Every point on the plane is described by two numbers.

What is an ordered pair?

An ordered pair is two numbers written as (x, y) that give the position of a point.

• First number: how far to move across, right if positive and left if negative.
• Second number: how far to move up or down, up if positive and down if negative.
• Order matters: (2, 1) and (1, 2) are different points.

What are the abscissa and the ordinate?

The abscissa is the x-coordinate, and the ordinate is the y-coordinate.

• Abscissa: the first number in the pair. In (2, 1) it is 2.
• Ordinate: the second number in the pair. In (2, 1) it is 1.

How do you plot a point on the coordinate plane?

1. Start at the origin (0, 0).
2. Move along the x-axis by the abscissa.
3. Move up or down by the ordinate.
4. Mark the point with a dot and label it.

What should you remember when graphing?

Does a point on a line make the equation true?

Yes, if a point lies on a line, its coordinates make the equation true.

Example: (2, 1) is on the line y = 2x – 3 because 1 = 2(2) – 3.

Counter-example: (3, 2) is not on the same line because 2(3) – 3 = 3, and 3 is not 2.

Why start with “y =”?

Start with “y =” because it makes the slope and y-intercept easy to read.

Example: To graph 6x + 2y = 8, rewrite the equation:

1. Subtract 6x from both sides: 2y = -6x + 8.
2. Divide every term by 2: y = -3x + 4.

Now graph the line using either the slope-intercept method or the table method.

How do you graph a linear equation by making a table?

A table of values lists x values and the matching y values that fit the equation.

1. Write the equation in “y =” form.
2. Choose at least three values of x, such as 0, 1, 2 and 3.
3. Put each x into the equation to find y.
4. Write each pair as (x, y).
5. Plot the points on the coordinate plane.
6. Join the points with a straight line using a ruler, and extend it past the points.

Worked example: graph y = 2x – 3

How do you graph a line using the slope and y-intercept?

This method uses m and b from y = mx + b to plot points without a table.

1. Write the equation in “y =” form.
2. Plot the y-intercept, (0, b), on the y-axis.
3. Write the slope as rise over run.
4. From the y-intercept, move up or down by the rise, then across by the run.
5. Mark the new point and join the points with a straight line.

Example: For y = -3x + 4, the y-intercept is 4 and the slope is -3.

• Plot (0, 4).
• Slope -3 means rise of -3 and run of 1, so move 3 down and 1 right to reach (1, 1).

What is the standard form of a linear equation?

The standard form of a linear equation is Ax + By = C, where A, B and C are numbers.

• Example: 6x + 2y = 8 is in standard form.
• To graph it: rewrite it as “y =” form first, here y = -3x + 4.
• Each form is useful: choose the form that suits the question.

How do you check if a point is on a line using a graph?

A point is on a line if the point sits exactly on the drawn line.

1. Graph the equation carefully.
2. Plot the point.
3. Check whether the point lies on the line.
4. Confirm by putting the coordinates into the equation. Both sides must be equal.

Evaluation

1. What is the slope of the line y = 3?
2. What can you say about the slope of the line x = -2?
3. Rewrite 6x + 2y = 8 in “y =” form.
4. What other names are given to the x-coordinate and the y-coordinate?
5. How can you tell from its coordinates whether a point lies on a line?

Assignment

Draw the graph of y = -3x + 4 and test a point on it, using these steps:

1. Make a table of values for x = 0, 1, 2 and 3.
2. Plot the points and join them with a straight line.
3. Check that the point (1, 1) lies on your line by putting its coordinates into the equation.